After the latest storms we started thinking about getting a power generator. One question I have is this: are there generators that are natural gas powered and that would be safe to install inside the house -- e.g. in the utility room -- and connected to the same gas line that feeds our water heater? How much space would I need for a unit like that? Thanks!

I would think a generator would fall under the same codes as a furnace. Some localities would probably not let you run an engine in anything but an enclosed. separate room. The real problem is going to be the exhaust gases. Since they won't be at low temperatures like a high efficiency furnace, you wouldn't be able to vent thru the wall and therefore need a chimney.

Possibly with venting you could do this.
I'd wonder about noise transmission, through the air and through the house framework.

I recommend 3 CO detectors in any case, and get the sensitive kind. 3 gives you 1 chance in 1 million of not detecting CO, at least on your purchase date. Dunno' how this reliability degrades over the life of the detectors.

After the latest storms we started thinking about getting a power generator. One question I have is this: are there generators that are natural gas powered and that would be safe to install inside the house -- e.g. in the utility room -- and connected to the same gas line that feeds our water heater? How much space would I need for a unit like that? Thanks!

Yes; they do make natural & LP gas powered generators.
No you can not install them inside a house.
Additionally, they are not very quiet.